Peculiar criticality of topological Hořava-Lifshitz black holes
We demonstrate the existence of criticality of the topological Hořava-Lifshitz(HL) black holes with a spherical horizon in the extended phase space. With the electric charge, we find that the critical behaviors of the HL black hole are nearly the same as those of van der Waals(VdW) system. For the uncharged case, the HL black hole has a peculiar criticality. The critical behavior is completely controlled by a parameter , but not the temperature . When is larger than a critical value , no matter what the temperature is, there will be the first-order phase transition. Moreover, we find that there is an infinite number of critical points which form a “ critical curve ”. As far as we know, this is the first time to find this kind of peculiar criticality.
Since the works of Hawking and BekensteinBekenstein-1973 (); Hawking-1975 (), thermodynamic properties of gravitational systems, especially black holes, have been extensively studied. Until now, there have been many approaches to calculating the temperature and entropy of black holes. Besides, the phase structure and critical phenomena of black holes have also been studied for quite a long timeHut.1977 (); Davies.1978 (); Sokolowski.1980 (); Pavon.1991 (); Lau.1994 (); Carlip.2003 (); Lundgren.2008 (). Since the pioneering work of Hawking and PageHawking.1983 () and the popularity of AdS/CFT correspondence, phase transitions of black holes in AdS space have aroused much interestChamblin.1999 (); Chamblin.1999b (); Cvetic.1999 (); Cvetic.1999b (); Peca.1999 (); Wu.2000 (); Biswas.2004 (); Myung.2005 (); Dey.2007 (); Myung.2008 (); Quevedo.2008 (); Cadoni.2010 (); Liu.2010 (); Sahay.2010 (); Banerjee.2011 (); Ma.2014 (); Ma.2014b ().
Recently, physicists reconsidered the critical behaviors of black holes in extended phase space, where the cosmological constant is treated as the thermodynamic pressure and its conjugate quantity as the thermodynamic volume of the black holes. It is shown that the criticality of some black holes in this extended phase space are very similar to those of a van der Waals (VdW) liquid/gas systemDolan.2011 (); Kubiznak.2012 (); Cai.2013 (); Chen.2013 (); Hendi.2013 (); Altamirano-2014 (); Mo.2014 (); Xu.2014 (); Ma.2015 (); Ma.2015b (); ZhaoHH.2015 (). Further, in this extended phase space other interesting critical behaviors such as reentrant phase transition, triple critical point, isolated critical point for several black holes have been exploredAltamirano.2013 (); Wei.2014 (); Dolan.2014 (); Frassino.2014 (). Thus, we want to know whether there are still other peculiar critical phenomena in the extended phase space. This is the first motivation of the present work.
Hořava-Lifshitz (HL) gravity, which was proposed by Hořava, is a power-counting renormalizable gravity theory and can be regarded as an ultraviolet complete candidate for general relativityHorava.2009 (). The HL gravity is constructed under two conditions: projectability and detailed balance. The breaking of projectability can simplify the theory. One has more freedom to construct the invariants under foliation of spacetime-preserving diffeomorphism. With the detailed balance, it is shown that the HL gravity has some problems, such as parity violating, ultraviolet instability and strong coupling problemCalcagni.2009 (); Charmousis.2009 (); Sotiriou.2009 (); Calcagni.2010 (). Therefore, it is also reasonable to abandon the detailed balance. Relaxing the projectability condition and deviating from detailed balance, some spherically symmetric black hole solutions has been derived in Lu.2009 (); Cai.2009c (). The thermodynamic quantities of some HL black holes are calculated in Cai.2009b (); Myung.2009 (); Kiritsis.2010 (); Liu.2014 (). In Cao.2011 () the authors studied the phase transition of a charged topological HL black hole with general dynamical parameter and the phase transition of a Kehagias-Sfetsos black hole. In Majhi.2012 (), the phase transition and scaling behavior of charged HL black holes with were studied. Through thermodynamic metrics, in Mo.2013 () the authors analyzed a unified phase transition of the charged topological HL black hole in the case of . And the same author concluded that no criticality exists in a topological charged HL black holeMo.2015 (). In the framework of horizon thermodynamics proposed by PadmanabhanPadmanabhan.2002 (), we also studied the HL black hole and found a universal phase structure and the criticalityMa.2017 ().
In this paper, we would study the criticality of topological HL black holes in the extended phase space under the more general consideration of nonzero (violation of the detailed-balance condition). We want to clarify the influence of the parameter on the thermodynamic properties and critical behaviors of the HL black holes. This is the second motivation of the present work. In the uncharged case, we find a peculiar criticality, for which there is an infinite number of critical points, in fact, a “critical curve”. For the charged HL black hole, there is the usual criticality similar to that of a VdW system. The parameter plays an important role in the generation of these critical phenomena.
The plan of this paper is as follows: In Sec.2 we introduce the HL black hole and the thermodynamic quantities. We also present the necessary demonstrations on some notations. In Sec.3 we analyze the conditions under which the temperature and the entropy of the HL black hole are positive. In Sec.4 we study the criticality of the HL black hole in the uncharged and charged case, respectively. In Sec.5 we summarize our results and discuss the possible future directions.
2 Thermodynamic quantities of Hořava-Lifshitz black hole
In this section, we simply introduce the Hořava-Lifshitz black hole and its thermodynamic quantities studied in Lu.2009 (); Cai.2009c (). The action of HL gravity without the detailed-balance condition is
where with the Cotton tensor. In this theory, there are several parameters: , , , , and . stands for the Lagrangian of other matter fields.
Compared with general relativity, there will be the relations for the parameters:
where and are Newton’s constant, the speed of light, and the cosmological constant respectively. In the following work, we will take the natural units: , so and .
Because only for the case with , general relativity can be recovered in the large distance approximation, we only consider in the following. In this case, from Eq.(2.2), one can see that must be negative.
corresponds to the so-called detailed-balance condition, under which HL gravity turns out to be intimately related to topological gravity in three dimensions and the geometry of the Cotton tensor. For the case with , HL gravity returns back to general relativity and the HL black hole becomes a Schwarzschild-(A)dS black hole. Therefore, we will consider the general values of in the region below.
For a static, spherically symmetric black hole, the metric ansatz can be written as
where denotes the line element for a two-dimensional Einstein space with constant scalar curvature . Without loss of generality, one can take (spherical/elliptic horizons), (flat horizons), and (hyperbolic horizons).
When a Maxwell field exists, it is shown that the metric function and is given byCai.2009c ()
where and and are integration constants. They are related to the black hole mass and electric charge:
where we have taken the natural units and set . The event horizon is defined through , where denotes the largest root of and its complete expression is given in the Appendix.
If we consider the cosmological constant as a variable and identify it with pressure , then according to Eq.(2.2), there is .
where is the position of the event horizon of the HL black hole.
According to the metric function, one can easily derive the temperature:
The entropy of the charged topological HL black hole is
Clearly, it is independent of the electric charge . The integration constant in the entropy cannot be fixed by some physical considerations. To determine , one has to invoke the quantum theory of gravity as argued in Cai.2009c (). For case, the area law of the black hole entropy is recovered if setting . Thus, for simplicity, we always set below (including the cases of ).
Other thermodynamic quantities can also be derived from the extended first law of black hole thermodynamics:
Here we do not consider the parameter as a thermodynamic variable. In the extended phase space, the black hole mass now should be considered as the enthalpy, . Due to the existence of the logarithmic term in the entropy, no Smarr-like relation for the HL black hole exists.
In this paper we are only concerned with the fixed charge ensemble. Thus, the Gibbs free energy is now defined by
by which we can analyze the global thermodynamic stability and phase transition of the HL black hole.
3 Positivity of entropy and temperature of Hořava-Lifshitz black holes
In this section, we will analyze the conditions under which the entropy and the temperature of the topological HL black holes are positive. The pressure is clearly positive according to the definition. According to Eq.(2.8), the entropy is a function of , and is independent of the electrical charge . After setting , the entropy is not always positive except for the case . To determine the positivity of , we fix the pressure and analyze the relations between and . In Fig.1, we show the relations between and for the positive entropy. We only pay attention to the case with . For , it is shown that the scope of becomes larger and larger as increases, and cannot reach to zero until . For , we find that the entropy is always positive so long as belongs to the region .
For the temperature, we only analyze the uncharged case for simplicity. For the charged case, the analysis is similar. According to Eq.(2.7), we can also depict the diagrams for positive temperature. For , the lower branch corresponds to zero temperature and the upper branch corresponds to the divergent temperature. Only on the right-hand side of the two branches is the temperature positive. For , the lower branch corresponds to the divergent temperature and the upper branch corresponds to the zero temperature. Outside the region enclosed by the two curves, the temperature is positive.
4 criticality of Hořava-Lifshitz black hole
According to Eq.(2.7), one can derive the pressure as a function of :
After a series expansion, it has the form
By comparing the above equation with the van der Waals equation, one can easily find the specific volume . So we would not introduce the specific volume, but directly study the behavior.
If the HL black hole has the similar criticality to that of a VdW system, the critical points should satisfy the following equations:
It is shown that the two equations are too complicated to be directly solved. However, we can employ the implicit differentiation on Eq.(2.7) to obtain
Requiring both , we can obtain
Substituting them into Eq.(2.7), we can obtain the interesting relations:
When , the equation of state reduces to
And Eq.(4.8) turns into
Excluding the case, it gives
In this case, the entropy and temperature are always positive. The equation of state has a simple form:
And Eq.(4.8) always holds. Obviously, in this case, no criticality exists.
In this case, the equation of state turns into
It can be easily checked that the equation with minus sign “-” has no real roots for . And the equation with plus sign “+” has the real roots:
According to Eq.(4), one can obtain
For any value of , there is the universal relation:
Therefore, in this case it is not a critical point, but a “critical curve”. As is shown in Fig.3, any point in this curve is the critical point. This critical behavior is not analogous to those of a VdW system or RN-AdS black hole or other black holes in AdS space, for which there is only one critical point.
The corresponding diagrams are depicted in Fig.4. The left plot of Fig.4 shows that the curves are exactly the same as the curves of a VdW liquid/gas system. However, there is an important difference. The critical behaviors of the uncharged HL black hole are controlled by the parameter , but not the temperature . The three curves in this plot have the same temperature. The values of determine the presence or absence of the criticality. When , there is an unstable region with a negative compression coefficient in the curve, and it should be replaced by a horizontal line determined by Maxwell’s equal area law. In this case, the smaller black hole/larger black hole phase transition occurs, which is reminiscent of the liquid/gas phase transition of the VdW system. When , no phase transition occurs. As is shown in the right plot of Fig.4, only if , the isotherms are critical ones regardless of the temperature. The different values of temperature only influence the position of the critical points.
In the left plot of Fig.4, one can notice another interesting behavior. When , the constant pressure line can intersect with the curves at three points, and when , the constant pressure line can intersect with these curves at five points. These points represent degenerate thermodynamic states with the same pressure and temperature. To ascertain which one of them is thermodynamically preferred, we need to calculate the Gibbs free energy. According to Fig.5, when , the diagrams exhibit “swallow tail” behavior, which is a typical feature of the first-order phase transition. We also find that at lower temperatures the HL black hole with a larger value of is more thermodynamically stable and at higher temperatures the HL black hole with a smaller value of is more thermodynamically stable. Considering the relations in Fig.6, it means that at a low temperature the smaller HL black hole is more thermodynamically stable and at a high temperature the larger HL black hole is more thermodynamically preferred.
In this case, Eq.(4.11) turns into
We find that the equation with “+” sign has no real roots and the equation with “-” sign has the same real roots as those in Eq.(4.15). However, in this case the critical pressure and the critical temperature are both negative,
In other words, there do not exist criticality and phase transition in this case.
According to Eq.(4.8), there should be the following requirement on :
And the existence of real roots requires or . For , the above inequality is clearly violated. Thus, no critical point exists in this case.
For , the constraints on are given in Table.1. According to Eq.(4), one can easily check that for the critical temperature and the critical pressure are negative when is in the regions in Table.1. Therefore, in this case there is also no physically acceptable critical behavior. Below we only focus on the case.
For , Eq.(4.8) has the solution (we always consider positive electric charges.). However, the corresponding critical pressure and critical temperature are both negative. Therefore, this case should be dropped. Thus, the solely feasible choice is . For the special value , the HL gravity will return back to the GR. Correspondingly, the HL black hole is reduced to the RN-AdS black hole. Therefore, in this case it certainly has criticality.
When situates in the interval , we can derive the critical point:
The universal ratio , which is independent of the electric charge . Because the interval for is so narrow, we should fine-tune the values of and to exhibit the criticality. Fig.7 and Fig.8 show the criticality and phase transition of the HL black holes with and . In general, the charged HL black hole for has the similar critical behaviors to that of RN-AdS black hole.
We can also calculate the critical exponents of the charged HL black hole. Introducing the following dimensionless quantities:
and replacing and in Eq.(4.1) with the new dimensionless parameters and and then expanding the equation near the critical point approximately, after chopping some very small numbers one can obtain
where and are all functions of . Eq.(4.23) has the same form as that for the van der Waals system and the RN-AdS black holeKubiznak.2012 (). Therefore, for this system the critical exponents should also be and . In addition, because , we also have the critical exponent . Obviously, they obey the scaling symmetry like the ordinary thermodynamic systems.
5 Conclusion and Discussion
We considered the HL gravity model with a nonzero . When the parameter increases from zero to one, the gravitational theory under consideration changes from the usual HL gravity to general relativity. Restricting , we studied the criticality of topological HL black hole in the extended phase space. We found that the HL black holes with the flat horizon ( and the hyperbolic horizon () have no criticality and phase transition. For the HL black hole with the spherical horizon (), with or without the electric charge, the criticality can appear. In the charged case, the HL black hole exhibits the similar critical behaviors and phase transition to that of a VdW liquid/gas system or RN-AdS black hole.
What is even more interesting is the uncharged HL black hole. In this case, we found some peculiar critical phenomena. First, there is an infinite number of critical points. The critical temperature and the critical pressure both depend on , while is free and is subjected to no constraint. Any value of corresponds to the critical position . So, we have a “critical curve”, but not only a critical point. Second, for the uncharged HL black hole it is the parameter that controls the criticality. This is very different from the criticality in the VdW liquid/gas system, where the temperature controls the critical behavior. We found that there is a critical , above which the smaller/larger black hole phase transition occurs. Especially, the criticality is completely determined by the parameter . By that we mean that only if , any temperature is the critical temperature.
We studied the HL black hole with . It is of great interest to extend our current study to a slightly more general . Another interesting future study would be to consider the phase transition and thermodynamic stability of the HL black holes in the non-extended phase space. It is also interesting to consider whether the peculiar criticality we found also exists in other black holes or thermodynamic systems.
Note added: After this work was published in PRD, the authors learned that the phenomena of critical curve have been found latelyHennigar.2017a (); Hennigar.2017b (); Dykaar.2017 (). So this peculiar criticality is of course not found by us for the first time. However, in our work there are some features that different from previous works. Our finding occurs for the black hole with spherical horizons in four-dimensional spacetime and the Lagrangian contains only quadratic powers of the curvature.
Appendix A The expression for the event horizon
M.-S. M would like to thank Professor Ren Zhao for illuminating conversations. This work is supported in part by the National Natural Science Foundation of China (Grants No.11605107 and No. 11475108) and by the Doctoral Sustentation Fund of Shanxi Datong University (2011-B-03).
- (1) J. D. Bekenstein, Black holes and entropy, Phys. Rev. D 7 (1973) 2333–2346.
- (2) S. W. Hawking, Particle creation by black holes, Commun Math Phys 43 (1975) 199–220.
- (3) P. Hut, Charged black holes and phase transitions, Mon. Not. R. astr. Soc. 180 (1977) 379–389.
- (4) P. C. W. Davies, Thermodynamics of black holes, Rep. Prog. Phys. 41 (1978) 1313–1355.
- (5) L. M. Sokolowski and P. Mazur, Second-order phase transitions in black-hole thermodynamics, J. Phys. A: Math. Gen. 13 (1980) 1113–1120.
- (6) D. Pavón, Phase transition in Reissner-Nordström black holes, Phys. Rev. D 43 (1991) 2495–2497.
- (7) Y. K. Lau, On the second order phase transition of a Reissner-Nordstrom black hole, Phys. Lett. A 186 (1994) 41–46.
- (8) S. Carlip and S. Vaidya, Phase transitions and critical behaviour for charged black holes, Class. Quantum Grav. 20 (2003) 3827–3837.
- (9) A. P. Lundgren, Charged black hole in a canonical ensemble, Phys. Rev. D 77 (2008) 044014.
- (10) S. W. Hawking and D. N. Page, Thermodynamics of black holes in anti-de Sitter space, Commun.Math. Phys. 87 (1983) 577–588.
- (11) A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, Charged AdS black holes and catastrophic holography, Phys. Rev. D 60 (1999) 064018.
- (12) A. Chamblin, R. Emparan, C. V. Johnson and R. C. Myers, Holography, thermodynamics, and fluctuations of charged AdS black holes, Phys. Rev. D 60 (1999) 104026.
- (13) M. Cvetic and S. S. Gubser, Phases of R-charged black holes, spinning branes and strongly coupled gauge theories, J. High Energ. Phys. 1999 (1999) 024.
- (14) M. Cvetic and S. S. Gubser, Thermodynamic stability and phases of general spinning branes, J. High Energ. Phys. 1999 (1999) 010.
- (15) C. S. Peça and J. P. S. Lemos, Thermodynamics of Reissner–Nordström–anti-de Sitter black holes in the grand canonical ensemble, Phys. Rev. D 59 (1999) 124007.
- (16) X. N. Wu, Multicritical phenomena of Reissner-Nordström anti–de Sitter black holes, Phys. Rev. D 62 (2000) 124023.
- (17) A. Biswas and S. Mukherji, On the Hawking–Page transition and the Cardy–Verlinde formula, Phys. Lett. B 578 (2004) 425–431.
- (18) Y. S. Myung, No Hawking–Page phase transition in three dimensions, Phys. Lett. B 624 (2005) 297–303.
- (19) T. K. Dey, S. Mukherji, S. Mukhopadhyay and S. Sarkar, Phase transitions in higher derivative gravity, J. High Energy Phys. 2007 (2007) 014.
- (20) Y. S. Myung, Y.-W. Kim and Y.-J. Park, Thermodynamics and phase transitions in the Born-Infeld-anti-de Sitter black holes, Phys. Rev. D 78 (2008) 084002.
- (21) H. Quevedo and A. Sánchez, Geometrothermodynamics of asymptotically Anti-de Sitter black holes, J. High Energ. Phys. 2008 (2008) 034.
- (22) M. Cadoni, G. D’Appollonio and P. Pani, Phase transitions between Reissner-Nordstrom and dilatonic black holes in 4D AdS spacetime, J. High Energ. Phys. 2010 (2010) 100.
- (23) H. Liu, H. Lü, M. Luo and K.-N. Shao, Thermodynamical metrics and black hole phase transitions, J. High Energ. Phys. 2010 (2010) 054.
- (24) A. Sahay, T. Sarkar and G. Sengupta, Thermodynamic geometry and phase transitions in Kerr-Newman-AdS black holes, J. High Energ. Phys. 2010 (2010) 118.
- (25) R. Banerjee, S. K. Modak and S. Samanta, Second order phase transition and thermodynamic geometry in Kerr-AdS black holes, Phys. Rev. D 84 (2011) 064024.
- (26) M.-S. Ma, F. Liu and R. Zhao, Continuous phase transition and critical behaviors of 3D black hole with torsion, Class. Quantum Grav. 31 (2014) 095001.
- (27) M.-S. Ma and R. Zhao, Phase transition and entropy spectrum of the BTZ black hole with torsion, Phys. Rev. D 89 (2014) 044005.
- (28) B. P. Dolan, Pressure and volume in the first law of black hole thermodynamics, Class. Quantum Grav. 28 (2011) 235017.
- (29) D. Kubizňák and R. B. Mann, P V criticality of charged AdS black holes, J. High Energ. Phys. 2012 (2012) 033.
- (30) R.-G. Cai, L.-M. Cao, L. Li and R.-Q. Yang, P-V criticality in the extended phase space of Gauss-Bonnet black holes in AdS space, J. High Energ. Phys. 2013 (2013) 005.
- (31) S.-B. Chen, X.-F. Liu and C.-Q. Liu, P-V Criticality of an AdS Black Hole in f(R) Gravity, Chinese Phys. Lett. 30 (2013) 060401.
- (32) S. H. Hendi and M. H. Vahidinia, Extended phase space thermodynamics and PV criticality of black holes with a nonlinear source, Phys. Rev. D 88 (2013) 084045.
- (33) N. Altamirano, D. Kubizňák, R. B. Mann and Z. Sherkatghanad, Thermodynamics of rotating black holes and black rings: Phase transitions and thermodynamic volume, Galaxies 2 (2014) 89.
- (34) J.-X. Mo and W.-B. Liu, P-V criticality of topological black holes in Lovelock–Born–Infeld gravity, Eur. Phys. J. C 74 (2014) 2836.
- (35) H. Xu, W. Xu and L. Zhao, Extended phase space thermodynamics for third-order Lovelock black holes in diverse dimensions, Eur. Phys. J. C 74 (2014) 3074.
- (36) M.-S. Ma and Y.-Q. Ma, Critical behaviors of a black hole in an asymptotically safe gravity with cosmological constant, Class. Quantum Grav. 32 (2015) 035024.
- (37) M.-S. Ma and R. Zhao, Stability of black holes based on horizon thermodynamics, Phys. Lett. B 751 (2015) 278–283.
- (38) H.-H. Zhao, L.-C. Zhang, M.-S. Ma and R. Zhao, Phase transition and Clapeyron equation of black holes in higher dimensional AdS spacetime, Class. Quantum Grav. 32 (2015) 145007.
- (39) N. Altamirano, D. Kubizňák and R. B. Mann, Reentrant phase transitions in rotating anti–de Sitter black holes, Phys. Rev. D 88 (2013) 101502.
- (40) S.-W. Wei and Y.-X. Liu, Triple points and phase diagrams in the extended phase space of charged Gauss-Bonnet black holes in AdS space, Phys. Rev. D 90 (2014) 044057.
- (41) B. P. Dolan, A. Kostouki, D. Kubizňák and R. B. Mann, Isolated critical point from Lovelock gravity, Class. Quantum Grav. 31 (2014) 242001.
- (42) A. M. Frassino, D. Kubizňák, R. B. Mann and F. Simovic, Multiple reentrant phase transitions and triple points in Lovelock thermodynamics, J. High Energ. Phys. 2014 (2014) 080.
- (43) P. Hořava, Quantum gravity at a Lifshitz point, Phys. Rev. D 79 (2009) 084008.
- (44) G. Calcagni, Cosmology of the Lifshitz universe, J. High Energy Phys. 2009 (2009) 112.
- (45) C. Charmousis, G. Niz, A. Padilla and P. M. Saffin, Strong coupling in Hořava gravity, J. High Energy Phys. 2009 (2009) 070.
- (46) T. P. Sotiriou, M. Visser and S. Weinfurtner, Phenomenologically viable Lorentz-violating quantum gravity, Phys. Rev. Lett. 102 (2009) 251601.
- (47) G. Calcagni, Detailed balance in Hořava-Lifshitz gravity, Phys. Rev. D 81 (2010) 044006.
- (48) H. Lu, J. Mei and C. N. Pope, Solutions to Hořava gravity, Phys. Rev. Lett. 103 (2009) 091301.
- (49) R.-G. Cai, L.-M. Cao and N. Ohta, Topological black holes in Hořava-Lifshitz gravity, Phys. Rev. D 80 (2009) 024003.
- (50) R.-G. Cai, L.-M. Cao and N. Ohta, Thermodynamics of black holes in Hořava–Lifshitz gravity, Phys. Lett. B 679 (2009) 504–509.
- (51) Y. S. Myung, Thermodynamics of black holes in the deformed Hořava–Lifshitz gravity, Phys. Lett. B 678 (2009) 127–130.
- (52) E. Kiritsis and G. Kofinas, On Hořava-Lifshitz “black holes”, J. High Energ. Phys. 2010 (2010) 122.
- (53) H.-S. Liu and H. Lü, Thermodynamics of Lifshitz black holes, J. High Energ. Phys. 2014 (2014) 071.
- (54) Q.-J. Cao, Y.-X. Chen and K.-N. Shao, Black hole phase transitions in Hořava-Lifshitz gravity, Phys. Rev. D 83 (2011) 064015.
- (55) B. R. Majhi and D. Roychowdhury, Phase transition and scaling behavior of topological charged black holes in Hořava–Lifshitz gravity, Class. Quantum Grav. 29 (2012) 245012.
- (56) J.-X. Mo, X.-X. Zeng, G.-Q. Li, X. Jiang and W.-B. Liu, A unified phase transition picture of the charged topological black hole in Hořava-Lifshitz gravity, J. High Energ. Phys. 2013 (2013) 056.
- (57) J.-X. Mo, No P–V criticality for charged topological black holes in Hořava-Lifshitz gravity, Astrophys Space Sci 356 (2015) 319–325.
- (58) T. Padmanabhan, Classical and quantum thermodynamics of horizons in spherically symmetric spacetimes, Class. Quantum Grav. 19 (2002) 5387–5408.
- (59) M.-S. Ma, R. Zhao and Y.-S. Liu, Phase transition and thermodynamic stability of topological black holes in Hořava-Lifshitz gravity, arXiv:1604.06998v2 .
- (60) R. A. Hennigar, R. B. Mann, and E. Tjoa, Superfluid Black Holes, Phys. Rev. Lett. 118 (2017) 021301.
- (61) R. A. Hennigar, E. Tjoa, and R. B. Mann, Thermodynamics of hairy black holes in Lovelock gravity, J. High Energ. Phys. 1702 (2017) 070.
- (62) H. Dykaar, R. A. Hennigar, and R. B. Mann, Hairy black holes in cubic quasi-topological gravity, J. High Energ. Phys. 1705 (2017) 045.